Exercise 14, § 1

Let L be a well-ordered set, E an A-module and $(E_\lambda)_{\lambda \in L}$ an increasing family of submodules of E such that: (1) there exists a $\lambda \in L$ such that $E = \{0\}$; (2) E is the union of the $E_\lambda$ for $\lambda \in L$; (3) if $\lambda \in L$ is such that the set of $\mu < \lambda$ admits a greatest element $\lambda'$, $E_\lambda$ is the direct sum $E_{\lambda'}$ and a submodule $F_\lambda$; (4) if $\lambda \in L$ is the least upper bound of the set of $\mu < \lambda$, then $E_\lambda$ is the union of the $E_\mu$ for $\mu < \lambda$. Under these conditions, show that E is the direct sum of the $F_\lambda$. (Prove by transfinite induction that each $E_\lambda$ is the direct sum of the $F_\mu$ such that $\mu \leq \lambda$.)